An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

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Section XVI - Unresolved Issues for the author

Summary for the Section :

• In this Section , we shall look at the 4 unresolved / outstanding issues that remain puzzling for the author :
  • the Factorial Functions ,
  • the Fourier Transform ,
  • the Reciprocals ,
  • the truncation errors in Taylor Series expansions .

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The Factorial Functions :

• In general , the { Factorial Functions } satisfy these 2 criteria :
  • 

    and :

  • 

  And it immediately follows that :

  • 

    and :

  • 

    yielding :

  • 

• While the factorial-type [ PI-of-(x) ] function does satisfy these criteria ,
  • there may be other functions that also qualify as { Factorial Function } .

  Vey simply , the reasons and full mechanics for Riemann's use of the [ PI-of-(x) ] function has not been fully understood .

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Fourier Transform :

• Fourier Series are originally set up for analysing { periodic functions } .

  And on the Fourier Transform :

  • the limits-of-integration are [ - infinity ] to [ + infinity ] on the forward-leg out , and

  • the limits-of-integration are [ - infinity ] to [ + infinity ] on the backward-leg in .

  Two (2) uncertainties are unresolved by the author here :

  • If we move forwards-and-backwards quite a few times here on the Transforms with these [ infinity-limits ] ,
    • do we loose something in error-terms in the longer-run ,

      or :

      are the Fourier Transforms indeed absolutely air-tight .

• A thought also being given here is this :
  • If we look at the Riemann's f(x) function up to a finite value of [ M ] ,
    • the number of [ steps / jumps ] are finite .

    Two (2) Degrees-Of-Freedoms ( DOF's ) will be associated with each of these [ step/ jump ] , namley :

    • the location of the [ step / jump ] within the relevent range ,
    • the size of the [ step / jump ] .

  Since the Degrees-Of-Freedoms ( DOF's ) are finite :

  • Would using { Discrete Transforms } here help at all ?

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Reciprocals :

• For our discussion here , let is set up the value [ XLN ] as an { Xtra-Large Number } .

  Then , reciprocals can be express via integrals in this format :

  • 

    yielding :

  • 

• Let us now sum up the [ reciprocals ] from [ 1 ] to [ V ] , and we have :
  • 

    yielding :

  • 

  And we note here that a prerequisite for the Integrals to exist is :

  • 

• Let us now { take-the-limit } as [ V ] goes to [ infinity ] , and we have :
  • 

    yielding :

  • 

  But let us recall that :

  • 

    so that , on expansion , the above term-on-the-very-right becomes :

  • 

  A quick comment here :

  • For the value of [ sigma = 0.5 ] and the value of [ V ] being at [ inifinty ] ,
    • the value of [ XLN ] should be at least [ infinity raised-to-the-4th-Power ] ,

      in order that the above term may go to [ zero ] .

    Is this a problem . Maybe

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Truncation Errors in the Taylor Series expansions :

• Riemann used the Taylor Series expansion for the [ logs ] and [ negative logs ] .

  But if we were to truncate the expansion at a very-small-but-finite value :

  • the number of terms remaining for the { prime number } [ 2 ] and

  • the number of terms remaining for the { prime number } [ 4409 ]

    might be significantly different .

• In short , are we treating all the { prime numbers } involved on a equal-and-fair basis .
  • May need to look at whether any bias is involved .

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go to the next section : Section XVII - Summarizing the 5 prime-related { Discrete Functions }

go to the last section : Section XV - Finite Values at [ s = 1 ] for a finite value of [ M ]

return to the Prime Numbers / L.C.M. / Zeta Functions HomePage

original dated 2009-5-08